Armature symmetrize wrong position
The symmetrize will use the armature's origin as the axis center, to fix your problem you could for example align the armature's origin to the dog's origin:


Make sure that dog's origin is centered to its geometry (Object > Set Origin > Origin To Geometry).
Select the dog object.
ShiftS > Cursor To Selected.
Select the armature object.
Object > Set Origin > Origin To 3D Cursor.
Go in Edit mode, symmetrize again.